A301425 Number of plane 5-regular simple connected graphs with 2n vertices.

1, 0, 1, 1, 6, 14, 98, 529, 4035, 31009, 252386, 2073769, 17277113

Offset: 6

Author

M. F. Hasler, Mar 20 2018

Keywords

• nonn
• more

Comments

We count here plane graphs, i.e., graphs embedded in the plane, up to embedding-preserving isomorphism, while such sequences as A003094 count planar graphs (counted up to abstract isomorphism). In this we follow the nomenclature of Brendan McKay, cf. link.

Examples

			There is only a(6) = 1 planar 5-regular simple connected graph with 2n = 12 vertices, which is the icosahedral graph, cf. MathWorld link. If we label the vertices 1, ..., 9, A, B, C, they are connected as follows: 1 -> {2 3 4 5 6}, 2 -> {1 6 7 8 3}, 3 -> {1 2 8 9 4}, 4 -> {1 3 9 A 5}, 5 -> {1 4 A B 6}, 6 -> {1 5 B 7 2 }, 7 -> {2 6 B C 8}, 8 -> {2 7 C 9 3}, 9 -> {3 8 C A 4}, A -> {4 9 C B 5}, B -> {5 A C 7 6}, C -> {7 B A 9 8}.
For other numbers of vertices, the number of plane 5-regular simple connected graphs is as follows:
14 vertices: 0  graphs,
16 vertices: 1  graph,
18 vertices: 1  graph,
20 vertices: 6  graphs,
22 vertices: 14  graphs,
24 vertices: 98  graphs,
26 vertices: 529  graphs,
28 vertices: 4035  graphs,
30 vertices: 31009  graphs,
32 vertices: 252386  graphs,
34 vertices: 2073769 graphs,
36 vertices: 17277113 graphs. (From the McKay web page.)
		

Links

• B. D. McKay, Plane graphs (see also the section on Planar Graphs on this page).
• Eric Weisstein's World of Mathematics, Icosahedral Graph.
• Index to sequences related to planar graphs

Crossrefs

Cf. A308489, A292109, A292972, A323281, A322917, A322929, A006821, A003094.